A necessary condition for punctured category to be balanced is "every mono is the kernel of some morphism" or dually "every epi is the cokernel of some morphism". In view of this, existence of counterexamples seems much more plausible than if one thinks of an 'almost abelian' setting. ---- And the map n:Z->Z form Eric Wofseys example is not the cokernel of something.
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Stephan MüllerJan 25 '13 at 11:33

3 Answers
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An additive category need not be balanced. Consider the full subcategory of abelian groups consisting of all torsion-free groups. Then for any $n\neq0$, the map $n:\mathbb{Z}\to\mathbb{Z}$ is monic and epic, but it is not an isomorphism unless $n=\pm1$. The only nontrivial part of this is that it is epic, and this is simply the statement that a map from $\mathbb{Z}$ to a torsion-free group is uniquely determined by its value at $n$.

By the way, this is the universal cocomplete symmetric monoidal linear category equipped with a line object $\mathcal{L}$ and an epimorphism $1 \to \mathcal{L}$ (namely the one above). When $1 \to \mathcal{L}$ is a regular epimorphism, it is already an isomorphism (compare this with $\mathbb{P}^0_{\mathbb{Z}}$).

A different example of an additive category which is not balanced (which is even quasi-abelian and has many injective objects) is the category of locally convex spaces
and continuous linear mappings. Every continuous bijection between two
locally convex spaces is mono and epi but not necessarily iso, take
e.g. the identity from an infinite dimensional Banach space to itself endowed
with the weak topology.